Death Spiral of electrons into the nucleus

Main Question or Discussion Point

So electrons were suppose to crash on nucleus according to classical physics. I want to understand the dynamics of it. The basic idea is that it will slow down as it emit photons. But with what force and mechanism?

The idea is that the constant rotation about the nucleus would cause a constant emission of electromagnetic waves. This emission would drain energy from the electron, its radius of gyration would slowly decrease until it drops into the nucleus.

Yeah so far its good, but I couldn't find any other force than Abraham-Lorentz force that could slow it down. And Abraham-Lorentz force is zero in this case since uniformly rotating particles have constant acceleration. (since the jerk is zero for constant acceleration)

Note that if you have a uniformly charged ring that rotates (constant current in a circular wire) there is no oscillating dipoles, and no electromagnetic radiation, just static electric and magnetic fields.

Thanks a lot. Do you happen to know how energy would be conserved if direction is also constant?

If the direction were constant, then there would be no acceleration, hence no emission of electromagnetic radiation, and conservation of energy would just keep the particle moving in a straight line. Of course the direction is not constant, because the Coulomb force between electron and nucleus is acting at right angles to the direction of travel, so necessarily changes the direction.

If the direction were constant, then there would be no acceleration, hence no emission of electromagnetic radiation, and conservation of energy would just keep the particle moving in a straight line. Of course the direction is not constant, because the Coulomb force between electron and nucleus is acting at right angles to the direction of travel, so necessarily changes the direction.

I mean what if the direction is constant with constant acceleration (if we consider a general case apart from the death spiral of electron). Then we would have radiation but no recoil force to balance energy conservation.

I mean what if the direction is constant with constant acceleration (if we consider a general case apart from the death spiral of electron). Then we would have radiation but no recoil force to balance energy conservation.

That is correct, but in that case the force producing the acceleration is doing work, so some external agency is adding energy to the system. A fairly prosaic example: a radio transmitter generates radio waves by moving charges back and forth in a straight line, and requires a power source to operate.

Circular motion is unusual in that the direction of the applied force is always perpendicular to the velocity so the centripetal force does no work. That's how the planets can stay in their orbits more or less forever without any energy being added to the system.

That is correct, but in that case the force producing the acceleration is doing work, so some external agency is adding energy to the system. A fairly prosaic example: a radio transmitter generates radio waves by moving charges back and forth in a straight line, and requires a power source to operate.

But if there is no recoil force, how does this energy is converted to electromagnetic energy? Because without no opposite force, all energy we supply should be converted to kinetic energy, what accounts for the radiated energy?

But if there is no recoil force, how does this energy is converted to electromagnetic energy? Because without no opposite force, all energy we supply should be converted to kinetic energy, what accounts for the radiated energy?

If you try pushing a charged particle through an electromagnetic field, you will feel an opposing force. That's how, among other things, electric motors work.

If you try pushing a charged particle through an electromagnetic field, you will feel an opposing force. That's how, among other things, electric motors work.

I think we have a misunderstanding. I mean

When there is nothing external, just one charge and a constant force upon it (doesn't matter what pulls it), Abraham-Lorentz force is 0(Force is constant, direction is constant, jerk is zero). But due to F=ma, it will radiate(non zero acceleration). And the work done on the charge will be considered as kinetic energy(simply 0.5mV2).

Total energy applied : just the kinetic energy of the particle
Total energy on the system: kinetic energy of the particle and its radiation.

What am I missing? There should be a resistive force to slow it down (converting applied energy into radiation), but according to Abraham-Lorentz force, there isn't. So how energy is conserved? Is there some other force, or is Abraham-Lorentz force is somehow nonzero?

No, that is not right. The work done on the charge, calculated by W=F⃗ ⋅d⃗ W=\vec{F}\cdot\vec{d}, will be equal to the sum of the electromagnetic energy radiated away and the change in the kinetic energy.

Yes that is what's suppose to happen. But if that is the case, we will have a smaller speed as we accounted for radiated energy. That means we had a smaller acceleration compared to a non-radiating case. If we had a smaller acceleration compared to the non-radiating case, by F=ma, we had a smaller equivalent force. Thus there must be a recoil force for constant acceleration-constant direction pulling. But according to Abraham-Lorentz force there is no recoil force (jerk is 0), so I couldn't find any force to balance this energy conservation.

Is it me misunderstanding Abraham-Lorentz force and it's indeed nonzero for nonzero jerk, or is there some another force slowing down the electron as it radiates?